Question: $\overline{AB} = \sqrt{113}$ $\overline{BC} = {?}$ $A$ $C$ $B$ $\sqrt{113}$ $?$ $ \sin( \angle ABC ) = \frac{7\sqrt{113} }{113}, \cos( \angle ABC ) = \frac{8\sqrt{113} }{113}, \tan( \angle ABC ) = \dfrac{7}{8}$
Solution: $\overline{AB}$ is the hypotenuse $\overline{BC}$ is adjacent to $\angle ABC$ SOH CAH TOA We know the hypotenuse and need to solve for the adjacent side so we can use the cos function (CAH) $ \cos( \angle ABC ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\overline{BC}}{\overline{AB}}= \frac{\overline{BC}}{\sqrt{113}} $ $ \overline{BC}=\sqrt{113} \cdot \cos( \angle ABC ) = \sqrt{113} \cdot \frac{8\sqrt{113} }{113} = 8$